We employ the proximal averaged Newton-type method for optimal control, for short PANOC, which was recently proposed in [1]}. PANOC performs fast Newton-type updates while an FBE-based linesearch endows it with global convergence properties while it uses the same oracle as the projected gradient method.

PANOC algorithm for problem (REF )[1]
\(\gamma \in (0,{1}{L_\ell })\) , \(L_\ell >0\) , \(\sigma \in (0,\tfrac{\gamma }{2}(1-\gamma L_\ell ))\) , \(u_0\in \Re ^{n}\) , \(x_0\in \Re ^{n_x}\) ,
L-BFGS memory length \(\mu \) .\(\nu =0,1,\ldots \) 

\(\bar{u}^\nu {}\leftarrow {}_{U}(u^\nu -\gamma \nabla \ell (u^\nu ) )\) 
\(r^\nu {}\leftarrow {}\gamma ^{-1}(u^\nu -\bar{u}^\nu )\) 
 \(d^\nu \leftarrow -H_\nu r^\nu \)  using L-BFGS \(u^{\nu +1}{}\leftarrow {}u^\nu - (1-\tau \!_\nu )\gamma r^\nu + \tau \!_\nu d^\nu \) ,
where \(\tau \!_\nu \)  is the largest number in \(\lbrace {1}{2^i}{}:{}i\in \rbrace \)  such that
\(\varphi _\gamma (u^{\nu +1}){}\le {}\varphi _\gamma (u^\nu )-\sigma \Vert r^\nu \Vert ^2\) 
